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# Stochastic Approach Examines Disease Emergence in a Seasonally Varying Environment

Effectively predicting disease dynamics and providing guidance for disease prevention and control are major goals in the public health sector. However, understanding population heterogeneity and environmental variability remains an ongoing challenge. Seasonal variations (e.g., temperature, rainfall, humidity, resource availability, and disease transmission pathways) cause fluctuations in factors like disease contact rates, human activity levels, viral growth, and death. In turn, these factors exert a strong influence on the occurrence of outbreaks of many infectious diseases, including cholera, Ebola, and influenza [3-5, 9]. Such outbreaks occur with much greater frequency during certain times of the year [7].

Figure 1. Transition probabilities $$r_i \Delta t + o (\Delta t)$$ for the branching process approximation. Figure courtesy of [1].
Cholera is a severe waterborne illness that is caused by the bacterium Vibrio cholerae. It is a seasonal disease in many endemic regions, and the peak values of infection usually occur annually during the rainy or monsoon periods. Although numerous ecological and biochemical studies have investigated the relationship between infectious diseases and seasonality, theoretical insight into cholera seasonality is scarce. A 2014 study considered a general ordinary differential equation (ODE) model of a cholera epidemic within a periodic environment, and demonstrated that the basic reproduction number $$(R_0)$$ of the ODE model serves as a disease threshold; the disease dies out when $$R_0 < 1$$ and persists when $$R_0 > 1$$ [10]. Yet the same outcome does not apply to a stochastic model, whose dynamics show that the disease may disappear even when $$R_0 > 1$$.

The lack of mathematical and statistical tools that account for seasonal variations in stochastic models limits researchers’ abilities to address important issues regarding environmental variability’s effects on the initiation and progression of disease outbreaks. In our stochastic investigation [1], we formulated a time-nonhomogeneous stochastic process that is based on the transition rates from the aforementioned ODE model; however, our analysis includes discrete random variables for susceptible $$(S)$$, infectious $$(I)$$, and recovered $$(R)$$ hosts, as well as bacteria $$(B)$$ in the environment. We extended existing approaches to obtain an analytic estimate for the probability of disease extinction when infection is introduced into a disease-free environment [2, 6, 8], then found this analytical estimate by linearizing the time-nonhomogeneous process and applying the backward Kolmogorov differential equations. We also derived approximations for the moments of the first extinction time. Using cholera as an example, we illustrated our stochastic method and expanded it to more general infectious disease models by applying theory from multitype branching processes.

Figure 2. Parameters $$\bar e$$, $$\bar s$$, and $$\bar h$$ are mean values and $$\epsilon_e$$, $$\epsilon_s$$, and $$\epsilon_h$$ are relative amplitudes for periodic functions $$\beta_E$$, $$\xi$$, $$\beta_H$$, where $$\beta_E (t) = \bar e \left(1 +\epsilon_e \sin\left(\frac{2 \pi t}{\omega}\right)\right)$$. Figure courtesy of [1].
For example, let us assume annual variation in direct $$(I$$ to $$S)$$ and indirect $$(B$$ to $$S)$$ transmission rates, as well as host viral shedding within the environment [1, 10]. Figures 1 and 2 provide the rates and parameters in the branching process, while Figure 3 shows that the 13 estimates from the numerical simulation of the time-nonhomogeneous process (blue circles) agree with the periodic branching process approximation. The estimate in Figure 3 confirms that if one infectious individual is introduced at $$\tau = 60$$ days (where $$\tau$$ is the initial time), then the probability of an outbreak is $$\cong 0.71$$ (with disease extinction at $$\cong 0.29$$). But if the individual is introduced later in the year at $$\tau = 212$$ days, then the probability of an outbreak is less than $$0.16$$. When only a few pathogens are present in the environment, the probability of an outbreak is much lower.

Unlike the earlier ODE model’s prediction, the stochastic model exhibits a positive probability for disease extinction when $$R_0 > 1$$. We applied an analytic estimate to compute the mean and standard deviation (SD) for the time until disease extinction when infection is introduced at 13 different instances: $$τ = 0$$, $$365/12$$, $$365/6$$, $$365/4$$…, $$365$$ (see Figure 4a). The mean time until disease extinction is three to four times greater if the infection is initiated with an infectious host, rather than with a single bacterium (per milliliter). In addition, the largest values for the mean and SD occur when the transmission and shedding rates are decreasing.

Figure 3. We use the branching process approximation to estimate the probability of an outbreak after one infectious individual or bacterium is introduced at time $$\tau$$ into a disease-free, seasonally varying environment. Figure courtesy of [1].

We checked the branching process estimate for the mean and SD for time until extinction against the time-nonhomogeneous stochastic process by simulating $$2 \times 10^4$$ sample paths, beginning at each of the 13 values of $$\tau$$ and continuing until we attained an outbreak level of $$I + 0.25B = 200$$. We recorded the times at which the sample paths hit zero and computed the mean and SD. Figure 4b plots the difference between the mean and SD based on the numerical simulations (NS) and analytical estimate (Analytic) from the branching process as a relative error, i.e., (NS-Analytic)/Analytic. The small relative errors indicate that the branching process agrees well with the time-nonhomogeneous process.

Figure 4. Mean and standard deviation (SD) of disease extinction time. 4a. Analytical branching process estimates for the mean and SD of time to disease extinction. The curves are means and the vertical bars are $$\pm$$ one SDs. 4b. The relative errors between the analytical estimates from the branching process and the numerical simulations from the full nonlinear nonhomogeneous process. The left panel displays the result when one infectious individual is initially introduced at time $$\tau$$ without the presence of the bacterium. The right panel depicts the result when a single bacterium is initially introduced at time $$\tau$$ in the absence of infectious hosts. Figure courtesy of [1].

Researchers could further extend these theoretical results to more general infectious disease models by applying multitype branching process theory. We hope that our stochastic modeling approach will offer additional insight into disease emergence within a seasonally varying environment.

Xueying Wang presented this research during a contributed presentation at the 2022 SIAM Conference on Mathematics of Planet Earth, which took place concurrently with the 2022 SIAM Annual Meeting in Pittsburgh, Pa., this July.

References
[1] Allen, L.J.S., & Wang, X. (2021). Stochastic models of infectious diseases in a periodic environment with application to cholera epidemics. J. Math. Biol., 82(6), 48.
[2] BacaeĢr, N., & Ait Dads, E.H. (2014). On the probability of extinction in a periodic environment. J. Math. Biol., 68(3), 533-548.
[3] Cohen, C., Kleynhans, J., Moyes, J., McMorrow, M.L., Treurnicht, F.K., Hellferscee, O., … PHIRST group. (2021). Asymptomatic transmission and high community burden of seasonal influenza in an urban and a rural community in South Africa, 2017-18 (PHIRST): A population cohort study. Lancet Glob. Health, 9(6), e863-e874.
[4] Dushoff, J., Plotkin, J.B., Levin, S.A., & Earn, D.J.D. (2004). Dynamical resonance can account for seasonality of influenza epidemics. Proc. Natl. Acad. Sci., 101(48), 16915-16916.
[5] Emch, M., Feldacker, C., Islam, M.S., & Ali, M. (2008). Seasonality of cholera from 1974 to 2005: A review of global patterns. Int. J. Health Geogr., 7, 31.
[6] Lahodny Jr., G.E., Gautam, R., & Ivanek, R. (2015). Estimating the probability of an extinction or major outbreak for an environmentally transmitted infectious disease. J. Biol. Dyn., 9, 128-155.
[7] Martinez, M.E. (2018). The calendar of epidemics: Seasonal cycles of infectious diseases. PLoS Pathog., 14(11), e1007327.
[8] Nipa, K.F., & Allen, L.J.S. (2021). The effect of demographic variability and periodic fluctuations on disease outbreaks in a vector-host epidemic model. In M.I. Teboh-Ewungkem & G.A. Ngwa (Eds.), The mathematics of planet earth: Vol. 7. Infectious diseases and our planet (pp. 15-35). Cham, Switzerland: Springer Nature.
[9] Nyakarahuka, L., Ayebare, S., Mosomtai, G., Kankya, C., Lutwama, J., Mwiine, F.N., & Skjerve, E. (2017). Ecological niche modeling for filoviruses: A risk map for Ebola and Marburg virus disease outbreaks in Uganda. PLoS Curr., 9.
[10] Posny, D., & Wang, J. (2014). Modelling cholera in periodic environments. J. Biol. Dyn., 8(1), 1-19.

Xueying Wang is an associate professor in the Department of Mathematics and Statistics at Washington State University. Her research areas lie in mathematical biology and dynamical systems with applications on transmission dynamics of infectious diseases and cluster solutions in weakly coupled neural networks. Linda J.S. Allen is a Paul W. Horn Distinguished Professor Emerita in the Department of Mathematics and Statistics at Texas Tech University. Her research interests include modeling infectious diseases in human, animal, and plant populations.

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